Topics
Units and Measurements
- Quantitative Science
- System of Units
- Derived Quantities and Units
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- Measurement of Length
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- Dimensions and Dimensional Analysis
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- Significant Figures
- Definitions of SI Units and Constants
Mathematical Methods
- Vector Analysis
- Scalar
- Vector
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- Vector Operations>Law of parallelogram of vectors
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- Concept of Calculus
- Differential Calculus
- Integral Calculus
Motion in a Plane
- Concept of Motion
- Rectilinear Motion
- Displacement
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- Key Parameters of Circular Motion
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Laws of Motion
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- Aristotle’s Fallacy
- Newton’s Laws of Motion
- Newton's First Law of Motion
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- Types of Forces>Fundamental Forces in Nature
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- Coefficient of Restitution e
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- Impulse of a Force
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- States of Equilibrium
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- Centre of Gravity
Gravitation
- Concept of Gravitation
- Kepler’s Laws
- Law of Orbit or Kepler's First Law
- Law of Areas or Kepler's Second Law
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- Newton's Universal Law of Gravitation
- Measurement of the Gravitational Constant (G)
- Acceleration Due to Gravity (Earth’s Gravitational Acceleration)
- Variation in the Acceleration>Variation in Gravity with Altitude
- Variation in the Acceleration>Variation in Gravity with Depth
- Variation in the Acceleration>Variation in Gravity with Latitude and Rotation of the Earth
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- Gravitational Potential Energy
- Expression for Gravitational Potential Energy
- Connection of Potential Energy Formula with mgh
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Mechanical Properties of Solids
- Mechanical Properties of Solids
- Elastic Behavior of Solids
- Stress and Strain
- Types of Stress and Corresponding Strain
- Hooke’s Law
- Elastic Modulus>Young’s Modulus
- Elastic Modulus>Bulk Modulus
- Elastic Modulus>Modulus of Rigidity
- Elastic Modulus>Poisson’s Ratio
- Stress-strain Curve
- Strain Energy
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- Friction in Solids
- Origin of Friction
- Types of Friction>Static Friction
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Thermal Properties of Matter
- Thermal Properties of Matter
- Temperature and Heat
- Measurement of Temperature
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- Ideal Gas Equation
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Sound
- Sound Waves
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Optics
- Fundamental Concepts of Light
- Nature of Light
- Ray Optics Or Geometrical Optics
- Cartesian Sign Convention
- Reflection>Reflection from a Plane Surface
- Reflection>Reflection from Curved Mirrors
- Total Internal Reflection
- Refraction of Light
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- Optical Instruments
- Simple Microscope or a Reading Glass
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Electrostatics
- Concept of Electrostatics
- Electric Charge
- Basic Properties of Electric Charge
- Additive Nature of Charge
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- Conservation of Charge
- Force between Charges
- Coulomb’s Law
- Scalar Form of Coulomb’s Law
- Relative Permittivity or Dielectric Constant
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- Electric Field
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Electric Current Through Conductors
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- Specific Resistance
- Variation of Resistance with Temperature
- Electromotive Force (emf)
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Magnetism
- Concept of Magnetism
- Magnetic Lines of Force
- The Bar Magnet
- Magnetic Field due to a Bar Magnet
- Magnetic Field Due to a Bar Magnet at an Arbitrary Point
- Gauss' Law of Magnetism
- The Earth’s Magnetism
Electromagnetic Waves and Communication System
- Foundations of Electromagnetic Theory
- EM Wave
- Sources of EM Waves
- Characteristics of EM Waves
- Electromagnetic Spectrum
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- Ground (surface) Wave
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- Sky wave propagation
- Communication System
- Elements of a Communication System
- Commonly Used Terms in Electronic Communication System
- Modulation
Semiconductors
- Concept of Semiconductors
- Electrical Conduction in Solids
- Band Theory of Solids
- Intrinsic Semiconductor
- Extrinsic Semiconductor
- n-type semiconductor
- p-type semiconductor
- Charge neutrality of extrinsic semiconductors
- p-n Junction
- A p-n Junction Diode
- Basics of Semiconductor Devices
- Applications of Semiconductors and P-n Junction Diode
- Thermistor
- Work Done by Variable Forces: The Power of Integration
- Dividing and Conquering (Integration)
- The Graphical Method: Area Under the Curve
- Example
Work Done by a Variable Force: The Power of Integration
The formula for work you already know is:
Crucial Limitation: This formula ONLY works when the force (F) is constant throughout the displacement (s).
Real-World Check: In many real situations, the force changes! We call this a Variable Force.
- If you launch a satellite very far from Earth, the gravitational force pulling it changes as its distance from Earth increases.
- When you stretch a spring, the force required increases the more you stretch it. This is a classic variable force!
Dividing and Conquering (Integration)
Since the force F is constantly changing, we can't use W = F × Δs because we don't know which value of F to use.
The brilliant method is to break the total displacement (s1 to s2) into a large number of infinitely tiny segments, called ds.
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Focus on a Tiny Segment (ds): This segment is so incredibly small that the force F over this distance is practically constant.
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Work for the Segment (dW): For this tiny segment, the work done (dW) is simply:
Work = Force × Tiny Displacement ⇒ dW = F . ds -
Total Work (W): To find the total work, we just add up (integrate) all the tiny dW's from the start point (s1) to the end point (s2).
W = \[\sum dW=\int_{s_{1}}^{s_{2}}F\cdot ds\]
The Graphical Method: Area Under the Curve
The total work done by a variable force is equal to the area under the Force-Displacement (F-s) graph between the initial and final displacements.
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The area of any curve segment is calculated by Height × Width. In our F-s graph, the area of a tiny strip is F × ds, which is exactly our definition of the tiny work dW. Summing these strips (integration) gives the total area, which is the total work, W.
- Linear Variation (Easy Case): If the force varies linearly (a straight line, like a spring), the area is a simple geometric shape (a trapezium or triangle). See Fig 4.1(b).
Fig 4.1 (b): Work done by linearly varying force.
-
Non-linear Variation (Calculus Case): If the force varies non-linearly (a curve), you must use integration (calculus) to find the area. See Fig. 4.1(a).
Fig 4.1 (a): Work done by a nonlinearly varying force.
Example
Example 4.4: Over a given region, a force (in newtons) varies as F = 3x2 - 2x + 1. An object is displaced from x1 = 20 cm to x2 = 40 cm by the given force. Calculate the amount of work done.
Solution:
Step 1: Check Units and Convert to SI (Crucial!):
The Force equation is in Newtons (SI unit), so Displacement must be in Meters (SI unit).
x1 = 20 cm = 0.2 m
x2 = 40 cm = 0.4 m
Step 2: Apply the Integration Formula:
W = \[\int_{x_1}^{x_2}F\cdot dx=\int_{0.2}^{0.4}(3x^2-2x+1)dx\]
Step 3: Perform the Integration:
W = \[\left[\frac{3x^3}{3}-\frac{2x^2}{2}+x\right]_{0.2}^{0.4}=[x^3-x^2+x]_{0.2}^{0.4}\]
Step 4: Substitute the Limits and Calculate:
W = [(0.4)3 - (0.4)2 + 0.4] - [(0.2)3 - (0.2)2 + 0.2]
W = [0.064 - 0.16 + 0.4] - [0.008 - 0.04 + 0.2]
W = [0.304] - [0.168]
W = 0.136 J
